Equidistribution of Small Points, Rational Dynamics, and Potential Theory
Annales de l'Institut Fourier, Volume 56 (2006) no. 3, p. 625-688

Given a rational function ϕ(T) on 1 of degree at least 2 with coefficients in a number field k, we show that for each place v of k, there is a unique probability measure μ ϕ,v on the Berkovich space Berk ,v 1 / v such that if {z n } is a sequence of points in 1 (k ¯) whose ϕ-canonical heights tend to zero, then the z n ’s and their Gal (k ¯/k)-conjugates are equidistributed with respect to μ ϕ,v .

The proof uses a polynomial lift F(x,y)=(F 1 (x,y),F 2 (x,y)) of ϕ to construct a two-variable Arakelov-Green’s function g ϕ,v (x,y) for each v. The measure μ ϕ,v is obtained by taking the Berkovich space Laplacian of g ϕ,v (x,y). The main ingredients in the proof are an energy minimization principle for g ϕ,v (x,y) and a formula for the homogeneous transfinite diameter of the v-adic filled Julia set K F,v v 2 for each place v.

Étant donné une fonction rationnelle de degré au moins 2 défini sur un corps de nombres k, nous montrons que pour chaque place v de k, il existe une seule mesure μ ϕ,v sur l’espace de Berkovich Berk ,v 1 / v tel que si {z n } est un séquence de points de 1 (k ¯) dont les hauteurs ϕ-canonique tendent vers zéro, alors les points z n et leurs Gal (k ¯/k)-conjugués sont équidistribués selon μ ϕ,v .

La preuve utilise un relèvement F(x,y)=(F 1 (x,y),F 2 (x,y)) de ϕ pour construire une fonction de Arakelov-Green g ϕ,v (x,y) de deux variables pour chaque v. La mesure μ ϕ,v s’obtient comme le laplacien (au sens d’espace de Berkovich) de g ϕ,v (x,y). Les ingrédients principaux de la preuve sont un principe de minimisation de l’énergie pour g ϕ,v (x,y) et une formule pour le diamètre transfini homogène de l’ensemble rempli de Julia v-adique K F,v v 2 pour chaque place v.

DOI : https://doi.org/10.5802/aif.2196
Classification:  11G50,  37F10,  31C15
Keywords: Canonical heights, rational dynamics, equidistribution, arithmetic dynamics, potential theory, capacity theory
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     author = {Baker, Matthew H. and Rumely, Robert},
     title = {Equidistribution of Small Points, Rational Dynamics, and Potential Theory},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {3},
     year = {2006},
     pages = {625-688},
     doi = {10.5802/aif.2196},
     mrnumber = {2244226},
     zbl = {pre05176555},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2006__56_3_625_0}
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Baker, Matthew H.; Rumely, Robert. Equidistribution of Small Points, Rational Dynamics, and Potential Theory. Annales de l'Institut Fourier, Volume 56 (2006) no. 3, pp. 625-688. doi : 10.5802/aif.2196. http://www.numdam.org/item/AIF_2006__56_3_625_0/

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